SPSS Data Analysis Examples: Poisson Regression



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SPSS Data Analysis Examples
Poisson Regression

Note: The examples on this page were done in SPSS 15. If you are using an earlier version of SPSS, you may need to use the genlog command.

Examples of Poisson Regression

Example 1. School administrators study the attendance behavior of high school juniors at two schools. Predictors of the number of days of absence include gender of the student and standardized test scores in math and language arts.

Description of the Data

Let's pursue Example 1 from above.

We have attendance data on 316 high school juniors from two urban high schools in the file poissonreg.sav . The response variable of interest is days absent, daysabs. The variables math and langarts give the standardized test scores for math and language arts respectively. The variable male is a binary indicator of student gender.

Let's look at the data.

GET
  FILE='D:\work\data\spss\poissonreg.sav'.

DESCRIPTIVES
  VARIABLES=male math langarts daysabs
  /STATISTICS=MEAN STDDEV VAR MIN MAX .

GRAPH
  /HISTOGRAM=daysabs .

FREQUENCIES
  VARIABLES=male.

Some Strategies You Might Be Tempted To Try

Before we show how you can analyze this with a Poisson regression analysis, let's consider some other methods that you might use.

OLS Regression - You could try to analyze these data using OLS regression. However, count data are highly non-normal and are not well estimated by OLS regression.
Negative Binomial Regression - Negative binomial regression does better with over dispersed data, i.e. variance much larger than the mean.
Zero-inflated Regression Model - This might be a good if there are more zeros than would be expected by either a Poisson or negative binomial model.

SPSS Poisson Regression Analysis

GENLIN
  daysabs WITH male langarts math
  /MODEL male langarts math
  INTERCEPT=YES
  DISTRIBUTION=POISSON
  LINK=LOG.

The output looks very much like the output from an OLS regression. The output begins the goodness of fit including log likelihood, AIC and BIC. These values can be used when comparing models.

Next comes the Tests of Model Effects. This section looks the same as the section of Parameter Estimates. This is because that we have entered all the variables as continuous variables. So each one of them has just one degree of freedom. With models where there are categorical predictor variables, this section will give the over effects of categorical variables and continuous variables as well.

The Parameter Estimates follows. You will find the Poisson regression coefficients for each of the variables along with standard errors, z-scores, p-values and 95% confidence intervals for the coefficients.

Now, just to be on the safe side, let's rerun the poisson command with the covb = robust option in order to obtain robust standard errors for the Poisson regression coefficients.

GENLIN
  daysabs WITH male langarts math
 /MODEL male langarts math
  INTERCEPT=YES
  DISTRIBUTION=POISSON
  LINK=LOG
 /CRITERIA METHOD=FISHER(1) SCALE=1 COVB=ROBUST.

Using the covb = robust option has resulted in a fairly large change in the model chi-square, which is now a Wald chi-square, based on log pseudo likelihoods, instead of a likelihood ratio chi-square. The robust standard errors attempt to adjust for heterogeneity in the model. The variable math was border-line significant without the covb = robust option and is clearly not significant with it.

Since math is not significant in the model with robust standard errors, we will rerun the model dropping that variable.

GENLIN
  daysabs WITH male langarts 
 /MODEL male langarts 
  INTERCEPT=YES
  DISTRIBUTION=POISSON
  LINK=LOG
 /CRITERIA METHOD=FISHER(1) SCALE=1 COVB=ROBUST.

Finally, we will use the emmean option to get the predicted value in days absent for male and female. In order to use the emmean option, we will have to specify variable male to be a categorical variable. The model specified this way is the same as the one above since male is a binary variable, except the reference group for male is now switched to male = 1. That is why the sign for the parameter coefficients are reversed.

GENLIN
  daysabs
  BY male
  WITH langarts
 /MODEL
  male langarts
  INTERCEPT=YES
  DISTRIBUTION=POISSON
  LINK=LOG
 /CRITERIA METHOD=FISHER(1) SCALE=1 COVB=ROBUST
 /EMMEANS TABLES=male SCALE=ORIGINAL.

Sample Write-Up of the Analysis

The poisson regression model predicting days absent from school stay from language arts and gender was statistically significant with likelihood ratio chi-square = 171.503, df=2 yielding p-value <.0001. The predictors langarts and male were each statically significant. For these data, the expected log count for a one-unit increase in language arts was -0.0146. This translates to a decrease of about 1.46 days absent for a one standard deviation increase in language arts when gender is held constant. Male students had an expected log count -0.41 less than female students which amounts to about 2.27 fewer days absent than females while holding language arts constant.

Cautions, Flies in the Ointment

It is not recommended that Poisson models be applied to small samples. What constitutes a small sample does not seem to be clearly defined in the literature.

SPSS Data Analysis Examples Poisson Regression